Stochastic Analysis

 

V4F1 - Summer semester 2017 

 

Thursday 16.15-18.00 and Thursday 12.15-14.00, Kleiner Hörsaal, Wegelerstr. 10 

Exercise sheets: Immanuel Zachhuber

Tutorial classes: Claudio Bellani / Monday 16-18, SemR 1.007

Exam: 1-3 August 2017 and 26-28 September 2017

 Topics

• SDEs: properties, weak solutions, transformations, stochastic flows, long time behaviour, large deviations, link with PDEs, variation of parameters, numerical schemes

Prerequisites 

Ito calculus for Brownian motion, see  e.g. Prof. Eberle's lecture notes on "Introduction to Stochastic Analysis" (pdf).

Lecture Notes

The first part of the course will be based on Prof. Eberle's lecture notes for Stochastic Analysis SS16 (pdf), in particular Chapters 2,3 but excluding processes with jumps. Some notes for material not covered by Prof. Eberle's lecture notes will be posted here:

• Note 1 : Stochastic differential equations : existence, uniqueness and martingale problems. (pdf) [version 1.1, posted 24/5/2016]
• Note 2 : Girsanov transform, Doob's h-transform. (pdf) [version 1.1, posted 24/5/2016]
• Note 3 : Brownian martingale representation theorem, Entropy and Girsanov transform, Boué-Dupuis formula, Large deviations. (pdf) [version 1.3, posted 16/6/2016]
• Note 4 : Kolmogorov theorem, Stochastic flows, Backward Ito formula. (pdf) [version 1.1, posted 29/6/2016]

Further References

• Rogers/Williams: Diffusions, Markov processes and martingales, Vol.2
• Bass: Stochastic processes
• Protter: Stochastic integration and differential equations

Problem sheets

• Sheet 1 (due on tuesday 2/5)
• Sheet 2 (due on tuesday 9/5)
• Sheet 3 (due on tuesday 16/5)
• Sheet 4 (due on tuesday 23/5)
• Sheet 5 (due on tuesday 13/6)
• Sheet 6 (due on thursday 22/6, based on the paper here)
• Sheet 7 (due on tuesday 27/6)
• Sheet 8 (due on tuesday 4/7)
• Sheet 9 (due on tuesday 11/7)
• Sheet 10 (due on tuesday 18/7)

Course Journal

• Lecture 18/4 : Overview of the course. Weak and strong solutions to SDEs, uniqueness, Yamada-Watanabe uniqueness theorem.
• Lecture 20/4 :  Ito-Doeblin formula. Martingale solutions to SDEs. Levy's characterisation of Brownian motion.
• Lecture 25/4 : Orthogonal infinitesimal transformations of Brownian motion, Bessel processes, Tanaka's example of an SDE with weak solution but not strong solution not pathwise uniqueness. Equivalence between martingale and weak solutions (to be finished).
• Lecture 27/4 : End of the proof of the equivalence between martingale and weak solutions (with invertible diffusion matrix). Time change of continuous local martingales. 
• Lecture 2/5 : Dubins-Schwartz theorem with proof. First example of change of time for SDEs. Knight's theorem.   General result about change of time for SDEs. 
• Lecture 4/5 : Example of non-uniqueness of weak solutions. One dimensional diffusions : scale function and speed function.
• Lecture 9/5 : Digression about the integral giving the change of time in the example of the previous lecture when alpha=1 as an application of Ito formula. Girsanov transform. Example in finite dimension. Change of probability on a filtered space, exponential martingales. Girsanov theorem for martingales. Example of the Brownian motion with drift.
• Lecture 11/5 : Discussion about quadratic variation and equivalent probability measures. Continuation of the digression on the change of time, proof that when alpha=1 the change of time is a.s. infinite. Relation with occupation measure and local time of Brownian motion. Construction of equivalent changes of measure, the Doob's h-transform. 
• Lecture 16/5 : Doob's h-transform continued. Diffusion bridges. Conditioning a diffusion not to leave a given domain. 
• Lecture 18/5 : Brownian motion conditioned to stay positive and other examples of conditioning.
• Lecture 23/5 : Continuation of the discussion of conditioned diffusions.
• Lecture 30/5 : Exponential tilting. Weak solution to SDEs via Girsanov theorem. Novikov criterion.
• Lecture 1/6 : Martingale representation theorem.
• Lecture 13/6 : Lecture on the methods of the paper: Beskos, Alexandros, Omiros Papaspiliopoulos, and Gareth O. Roberts. "Retrospective exact simulation of diffusion sample paths with applications.'' Bernoulli (2006): 1077-1098. (by I. Zachhuber)
• Lecture 20/6 : Proof of the MRT. Variational properties of Girsanov transform. Relative entropy.
• Lecture 22/6 : Follmer's drift, Boué-Dupuis theorem. 
• Lecture 27/6 : Applications of Boué-Dupuis : concentration of measure, large deviations.
• Lecture 29/6 : (done the 27/6)
• Lecture 4/6 : Stratonovich integration. Doss-Sussmann method.
• Lecture 6/7 : Doss-Sussmann method in more than one dimension. Brownian motion on a manifold.
• Lecture 11/7 : Stochastic Taylor expansion. Strong and weak error. Numerical methods for SDEs. Analysis of the Euler scheme.
• Lecture 13/7 :
• Lecture 18/7 :
• Lecture 20/7 :